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Here's the full exchange that Dowd had with George Stephanopoulos:

STEPHANOPOLOUS: I do want to bring that to Matthew Dowd. I gotta say I was a little bit surprised, Matthew, at that first round where Governor Romney at first did not want to draw any distinctions with Newt Gingrich. And as David Muir pointed out, in that first exchange on who's the more consistent conservative, it was actually Newt who was more on offense and everybody expected him to be on defense tonight.

DOWD: Yeah, that was the amazing thing, 'cause everybody predicted, and even Newt had signaled, Newt Gingrich had signaled that he was gonna deflect the attacks and not engage on 'em. There was not a single attack tonight that he did not not engage on. He engaged and responded to every single one. He was right up front.

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because he clarifies the point by saying, "He engaged and responded to every single one." So how did the extra not creep in? It wasn't a case of stammering or self-repair, because he says "not not" emphatically, with stress on the second not and a fall-rise intonation contour. The fall-rise contour can be heard in other more comprehensible instances of "not not," as in those discussed by Laurence R. Horn as "double particle negation" in his 2010 book The Expression of Negation (pp. 118-9, viewable on Google Books). In these cases, Horn says, "a doubly negated predicate may be primed by the previous ascription of the corresponding simple predicate by the speaker or an interlocutor":

Bart: Dad, are you licking toads?
Homer: I'm not NOT licking toads.
(Exchange from 2000 episode from "The Simpsons")

Lucy: Are you friends with Mary?
Robbie: I'm not NOT friends with her.
(Dialogue from ABC TV family comedy "Seventh Heaven", 2001)

Horn writes that in such examples "the double negatives may fail to completely cancel out, instead amounting to a weaker positive than their target would have provided":

Evidently being 'not not friends' with someone is weaker than being friends with her, while 'not not licking' the putatively red toads of Micronasia [sic] is more innocent than actually licking them. The DN coerces a reading of 'not friends' and 'not licking' as virtual contraries rather than pure contradictories, so that the normally excluded middle is explicitly unexcluded.

But I have a hard time figuring out exactly how Dowd ended up using this type of doubly negated predicate in the relative clause, "that he did not not engage on." There was, one could argue, some priming of the "corresponding simple predicate" in his exchange with Stephanopoulos: they were talking about the debate-watchers' assumption that Gingrich would not engage on the attacks, so Dowd is turning that around to say that he did engage on all of them. But what is gained by trying to say that he "did not not engage on all of them"? It doesn't seem to correspond well to the examples Horn gives where a "weaker positive" is needed.

I think that Dowd may have had in mind a double-negative construction along the lines of:

Everybody predicted he would not engage on the attacks.
It turns out he could not not engage on the attacks (every single one of them).

Horn has written about the "could not not" (or "couldn't not") style of double negation, going back to his 1978 article, "Some Aspects of Negation." He gives the example sentence:

You can't not go = You must go, You have to go.

and says that this example, "with one negation each outside and within the scope of a possibility (or permission) operator yielding the A-value of the appropriate type, represents a paradigm case of double negation." He goes on to discuss equivalents in other languages (further detailed in Chapter 5 of Horn's 1989 book, A Natural History of Negation).

So my best guess for what was going on in Dowd's brain is that he was conflicted between saying

There was not a single attack tonight that he did not engage on.

and

He could not not engage on every single attack.

But instead of saying either one of these two-not statements, he somehow blended them together into a three-not statement. It just goes to show how our nots can tie us up in knots.

Kenny said,

I suppose we're supposed to leave this issue aside, but the triple negative is so thoroughly covered I don't think there's anything else to say. Is a verb-particle construction the same as what some call a phrasal verb? If so, I don't think "engage on" qualifies.

I think this "on" is topic "on". I noticed it with a lot of my political-science-major friends. They use it pretty much anywhere you can use "about" or "with respect/regard to". It is my understanding that in a phrasal-verb construction, the particle and the verb cannot be split, but I think "engage on" is perfectly amenable to splitting. I don't know that much about the stances of the Republican candidates, so I might not have chosen proper/realistic topics for the following examples.

Ran Ari-Gur said,

Ran Ari-Gur said,

It seems like what Dowd wanted was "There was not a single attack tonight that he did not engage on", with emphasis on the not; that is, I think he wanted the prosody and intonation of the double negative construction, even though logically only a single negative made sense.

I seem to recall that Dr. Liberman has written on Language log about constructions such as "to miss not ___ing …" (meaning "to miss ___ing …"), and quoted Larry Horn as pointing out that although "miss" has some negative sense, it doesn't license (some?) NPIs, with the result that something like "he missed not seeing anyone" can't simply be fixed to *"he missed seeing anyone". (That's my own example — I don't remember Dr. Horn's — but I believe I have the idea right.) I think this situation may be similar; Dowd may have felt that the first "not" was necessary in order to license the double-negative-style intonation of the second.